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Relating to my other question: Modules over Hopf Algebras and $E_2$-algebras

Preliminary: Let $A$ be an associative dg-algebra that is also an algebra over the brace operad. Let $M$ and $N$ be left dg-modules and assume we have previously constructed a quasi iso.

$$ f_k:A^{\otimes k}\to A^{op} $$ from the dg-algebra structure of $A$ to its opposite algebra based on the brace structure (which makes $A$ homotopy commutative).

We can construct the chain complex (ignore signs for now)

$$ M \otimes N:= M\otimes_k \mathfrak B (A) \otimes_k N $$ $$ \partial (m\otimes a_1 \otimes \cdots \otimes a_n \otimes n):= dm\otimes \cdots \otimes a_n \otimes n +m\otimes d(a_1)\otimes \cdots \otimes n + \cdots + $$ $$ m \otimes \cdots \otimes a_n \otimes dn + $$ $$ a_1\cdot m\otimes a_2 \otimes \cdots \otimes n + f_2(a_1,a_2)\cdot m \otimes \cdots \otimes n+ \cdots +f_n(a_1 ,\ldots ,a_n)\cdot m\otimes n + $$ $$ m\otimes a_1 a_2 \otimes \cdots \otimes n + \cdots + m \otimes a_1 \otimes \cdots \otimes a_{n-1} \otimes a_n \cdot n $$

where $\mathfrak B (A)$ is the bar complex of $A$. This is my attempt to turn the category of left modules over $A$ into a monoidal (or maybe monoidal up to homotopy) category.

The problem then becomes that this new chain complex is not a left module, at least with the logical choice of action on the m component of the tensor. The action of $A$ on the first component does not respect the differential of $M\otimes N$ and so we don't get a left dg-module out.

Question1: Is there a way to change this construction to ensure that what we get out is a true left dg-module over $A$?

Question2: If $M$ and $N$ were instead left $A_{\infty}$ modules over $A$, can we (easily?) alter the construction to make the result an $A_{\infty}$ module?

Question3: If we collapse everything by taking cohomology, then will the result be a left module over $H^*(A)$ (which is now commutative). Do we just get Tor back? Is there any way to get anything more than just classical results about commutative algebra by doing this?

Thanks, -Matt

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